I have recently had the rare pleasure of having lengthy, one-on-one pow-wows with the most popular man in English football and probably the most divisive. When I tell you the latter person isn't Luis Suárez, then I imagine you'd quickly, and correctly, guess the Chelsea manager José Mourinho. His antithesis? The always smiling, relentlessly amenable Everton manager, Roberto Martínez.

You'd expect them to be very different personalities, and they are. On the way to meet Martínez, the taxi driver – a Liverpool fan – noted approvingly that he'd had cooks and cleaners from the training ground in his car and they all said what a cheerier place Everton was these days. "Much better than under that grumpy bugger before," he said. I was speaking to Martínez about the data revolution in football and for more than an hour he was engaged and thoughtful; this despite the fact that he ostensibly disagreed with the entire premise of the article: that football, too, can be moneyball-ed.

Mourinho, whom I interviewed for this month's Esquire magazine, has no interest in putting anyone at ease. There's a story that when he addresses his players he stands with his back to the sun, so they have to squint and concentrate harder on what he's saying – which is probably apocryphal, but you'd like to be true. Mourinho arrived on the dot for our meeting, was formidably sharp-witted, and made it obvious that he didn't care what anyone – especially me – thought about him. Which, perversely, made me quite like him.

These men were most disparate, however, when they talked about football. Martínez's favourite word is "philosophy": he loves to talk about how the game is played. Mourinho is much more interested in "methodology": how best to prepare his teams for success. Both of them believe their approach is the most effective but their differences are revealing. It would perhaps be more accurate to say that Martínez is the most romantic man in English football and Mourinho the most pragmatic.

Martínez, a Catalan, arrived in England aged 22 with two Spanish chums, new signings of Wigan Athletic. "A big, big shock: you couldn't find an espresso anywhere," he told me, still clearly traumatised. "I'm not saying ham or olive oil, that's in a delicatessen. But in 1995, you couldn't have a coffee anywhere. Five o'clock everything would shut – we'd go for our siesta, get up, go out and everything was finished and done."

Worse for Martínez, an elegant midfielder, were his instructions on the pitch. "I couldn't tackle," he said. "But managers were asking me to tackle. I'd say: 'Why don't you ask me to pass the ball 20 yards, which nobody else can do?' And the other guy who was great at tackling, he was asked to pass the ball. I became average at tackling, but then I wasn't playing to my strengths, so I became average at everything."

The experience would become a core tenet of his management style: "I will never ask a player to do something he's not good at."

Listening to Martínez speak, I couldn't help picture another Spanish playmaker, Juan Mata. Mata, Chelsea's best player for the past two seasons, fell out of favour with Mourinho ostensibly because he didn't contribute enough to the team off the ball. Mourinho decided Mata was a luxury he could live without. Martínez, you'd imagine, would have built his squad around him.

When he talked about transfers, Martínez said it was essential for him to "fall in love with a player". Mourinho may not entirely agree on that, either. His two blowouts since returning to Chelsea have been the Duracell bunny Willian for £32m and the £21m slab of Balkan granite Nemanja Matic. Both have been instrumental in putting Chelsea clear at the top of the table, but these are not lust objects, they are practical buys. Less a Bose sound system, more a nice extractor fan.

The recurring complaints about Mourinho are that he is egotistical and his teams play boring football. Both underestimate him. In sport there are examples of individuals and teams that combine sainted grace with utter dominance – one thinks of Roger Federer, for a time, and Barcelona – but they are outnumbered by the unemotional machines. Mourinho accepts this. That's why, when he called Arsène Wenger "a specialist in failure", it must have stung the Arsenal manager. It was, in a sense, objectively undeniable.

Mourinho, unencumbered by a stellar career as a player in professional football, is completely hard-headed about what his teams need to win. A squad without a reliable centre‑forward, as he inherited at Chelsea, must be rock-solid in every other area. Mourinho denies that he favours a particular style of player – the demands of each club are unique – but he did once have an alsatian named Gullit: "Attacking, creative player," he has said, "but at the same time with fitness, power, mental strength."

Does Mourinho have a philosophy? "I say and say, and I repeat and I repeat: 'The group is more important than the individual,'" he told me. "So my initial job is to make the players work for the team, to work for each other, to achieve their individual objectives as a consequence of teamwork."

Martínez curates teams that are often a pleasure to watch. "I've got a way of playing," he accepts. "I'll never give you the ball, I want to have most of the possession." His aim is a romantic one: he would like the clubs he manages – even Wigan – to play like Barcelona at their peak. "The sign of a good team is when everyone knows what you are doing but they can't stop you."

Any neutral fan would will Martínez to succeed, and he is certainly shrewd enough to pull it off. But Mourinho's record over the years tells its own story. If Martínez creates a world of football as we'd all like it to be, then Mourinho plays in the world as it is.